Thin Magneic Layer

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The Astrophysical Journal, 670:221-230, 2007 November 20 2007. The American Astronomical Society. All rights reserved. Printed in U.S.A.

Bubble Wrap

for

Bullets:

The

Stability

Imparted

by

a

Thin

Magnetic Layer

L. J. Dursi Canadian Institute for Theoretical Astrophysics, University of Toronto, Toronto, ON,

M5S 3H8, Canada; [email protected]

Received 2007 June 24; accepted 2007 July 30

ABSTRACT

There has been significant recent work by several authors which examines a situation where athin magnetic layer is "draped" over a core merging into a larger cluster; the same process also

appears to be at work in a bubble rising from the cluster center. Such a thin magnetic layercould thermally isolate the core from the cluster medium, but only if the same shear processwhich generates the layer does not later disrupt it. On the other hand, if the magnetized layercan stabilize against the shear instabilities, then the magnetic layer can have the additionaldynamical effect of reducing the shear-driven mixing of the core's material during the merger

process. These arguments could apply equally well to underdense cluster bubbles, whichwould be even more prone to disruption. While it is well known that magnetic fields cansuppress instabilities, it is less clear that a thin layer can suppress instabilities on scalessignificantly larger than its thickness. We consider here the stability imparted by a thin magnetized layer. We investigate this question in the most favorable case, that of twodimensions, where the magnetic field can most strongly affect the stability. We find that inthis case such a layer can have a significant stabilizing effect even on modes with wavelengths

much larger than the thickness of the layer l . To stabilize modes with 10 l requires only

that the Alfvn speed in the magnetized layer is comparable to or greater than the relevant

destabilizing velocity the shear velocity in the case of pure Kelvin-Helmholtz likeinstability or a typical buoyancy velocity in the case of pure Rayleigh-Taylor like instability.We confirm our calculations with two-dimensional numerical experiments using the Athenacode.

Subject headings: galaxies: clusters: general ; hydrodynamics ; instabilities ; MHD ; X-rays:galaxies: clusters

1. INTRODUCTION

It has been known for some decades in the space science community that an object movingsuper-Alfvnically in a magnetized medium can very rapidly sweep up a significant magneticlayer which is then "draped" over the projectile (e.g., Bernikov & Semenov 1979 ). Afterrecent observations of very sharp "cold fronts" in galaxy clusters (see, for instance,Markevitch & Vikhlinin 2007 ), there has been significant interest in applying this idea of

magnetic draping in galaxy clusters (e.g., Vikhlinin et

al. 2001 ; Lyutikov 2006 ;

Asai et al.

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2004 , 2005 , 2006 ), as such a magnetic field could inhibit thermal conduction across the front(e.g., Ettori & Fabian 2000 ), allowing it to remain sharp over dynamically long times.

The effect of a strong draped magnetic layer could be even greater for underdense objects,such as for bubbles moving through the intercluster medium, as seen in many cool-core

clusters (e.g., McNamara et al. 2005 ; Brzan et

al. 2004 ). In this

case, the bubble would

be

quickly disrupted on rising, absent some sort of support (e.g., Robinson et al. 2004 ). However,

the draping of a pre-existing magnetic field may strongly alter the dynamics, as seen recentlyin simulations (Ruszkowski et al. 2007b , 2007a ).

However, the same shear motion which gives rise to the magnetic draping can also drive

instabilities which could then disrupt the layer. It is clear that magnetic fields can stabilizeagainst shear instabilities (Chandrasekhar 1981 ), and this was considered in this context in

Vikhlinin et al. (2001 ). However, in that work, the instability was considered between twosemi-infinite slabs with differing magnetic fields that is, the geometric thinness of themagnetic field was not taken into account. This is an appropriate regime for considering

perturbations much smaller than the thickness of the layer, but less so for modes which could disrupt the layer and contribute to the stripping of the core. For these modes, clearly that thelayer is in fact thin must have some effect, and naively, one might expect that the thin layerwould only be effective in stabilizing modes with sizes comparable to the breadth of the layer.

Here we consider the linear growth of Kelvin-Helmholtz and Rayleigh-Taylor instabilitiesin the presence of a thin magnetized layer. We consider the instability in two dimensions, withfield lines lying in the plane and parallel to the interface, and with perturbations along thedirection of the magnetic field. This is the case in which the layer could most stronglyinfluence the stability; if such a thin layer could not stabilize the flow in such a restrictedgeometry, it would surely be torn up by instabilities in the more realistic three-dimensional

case. In three dimensions, interchange modes with wavenumbers perpendicular to the field

lines are essentially unaffected by the presence of the magnetic field, and thus, the fieldcannot stabilize these modes. However, even in this case, overall mixing can be reduced bythe presence of a quite weak field (e.g., Gardiner & Stone 2007 ), much weaker than expectedin this case, and the introduction of such a stark asymmetry (modes in one plane attenuated

but in a perpendicular plane unaffected) could have interesting observable consequences.

In 2 we pose the problem, in 3 and 4 we derive the growth rates and stability boundaries for the Rayleigh-Taylor and Kelvin-Helmholtz instabilities, respectively, and in 5 we confirm our analytic results with two-dimensional numerical experiments using the

Athena code (Gardiner &

Stone 2005 ). We conclude

in 6 .

2. LINEAR THEORY

We follow the approach and notation of Chandrasekhar (1981 ), particularly from his 105.We begin with the equations of two-dimensional, incompressible, inviscid

magnetohydrodynamics,

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where xi is the i-coordinate and can take the values ( x, z ), U i

is the velocity in the i-coordinatedirection, is density, is any fluctuation in the density, p is pressure, B is the magnetic field,

g is the gravitational acceleration with gravity pointing "down" (e.g., in the direction of - ),

and summation over repeated

indices is implied. We

consider velocity, magnetic fields,

and pressure of the form

where B, W , and are constant within each region. We assume that all velocities are highlynonrelativistic, so that displacement currents and relativistic effects may be neglected. Wealso assume that the flow velocities are much less than the sound speed and, thus, mayconsider incompressible flow. A sketch of the situation under consideration is shown inFigure 1 .

(73 kB)

Fig. 1 Sketch of the problem under consideration. We consider a three-layer problem,with two semi-infinite, uniform, magnetic field free regions separated by a layer with ahorizontal magnetic field. We consider only field (and disturbances) in the direction ofthe shear. We generally consider the magnetized layer to have the same density andvelocity as the top fluid. Disturbances are modeled as periodic in the horizontaldirection and vanishing at in the vertical direction.

Following Chandrasekhar

(1981 , 105), we

consider plane wave perturbations

of the formei(kx+nt ) for b x, b z , u, w, dp, and . Linearizing the equations above results in

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Because we allow the magnetic field strength to vary between layers, there are extra terms inequations (7) , (8) , and (9) proportional to B which do not vanish. Within the layers

themselves, these terms are identically zero, but their existence will lead to more complex boundary conditions at the interfaces.

2.1. Solution within Uniform Layers

Equations (9) , (10) , (11) , and (12) can be solved to express b x, b z , b , b , , and u in

terms of w and its derivatives, the system parameters B, B , , , and U , the growth rate n, andthe wavelength of the disturbance k . This means that we can begin to write the dispersionrelation n(k ) in terms of only the perturbed z -velocity w and the known parameters. We can dothis by using equations (7) and (8) to eliminate dp and then eliminate all other perturbedvariables in favor of w, leaving

Within the three layers, where B = 0, U = 0, and = 0, we have

which since we are only interested in negative n2, leaves us with only the solutions

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as in the case of a uniform field and as anticipated in Figure 1 .

2.2. Boundary Conditions between Layers

The normal displacement of the interface is -iw/(n + kU ); this can be seen by expressing the perturbed interface position z in the same form as the other perturbed variables, and thelinear-order evolution equation for z becomes i(n + kU ) z = w.

Because the displacement of the interface must be unique, w/(n + kU ) must be continuousacross the interfaces. This and a related quantity, the shifted time derivative ( n + kU ), occurfrequently enough that it is useful to express equation (14) in terms of these quantities. Doingso results in

Integrating this equation over an infinitesimal region across either of the interfaces gives usthe boundary conditions across this interface. In doing so, terms that contain no derivativesvanish in the limit, and we are left with

where [ f ] indicates the jump in a quantity f across the interface and the subscript i refers to thevalue at the interface. Note that when B is constant, this reduces exactly to the homogeneousfield case found in Chandrasekhar (1981 , 106).

2.3. Matching the Solutions

In the top and bottom layer, one of the two solution branches [ w exp(+ kz ) and w exp(-kz ), respectively] is clearly unphysical. In the middle layer, however, both can coexist, and so

we have as forms for the solutions

Because w/(n + kU ) must be continuous across the interface, we have

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These can be solved for the components of the intermediate velocity in terms of the outer layer

velocities, giving

We now have two boundary conditions to satisfy equation (18) at the two interfaces between the layers. The top interface condition gives us

and the bottom interface gives us

This gives us two equations in terms of w1 and w3. Using our expressions for w2 and then

solving the first equation for w3 in terms of w1 and substituting

the result into the second givesus our final dispersion relation of

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where the Atwood number, At, is defined to be the nondimensional density difference

It is worth noting that in the no magnetization limit vA 0, the only term containing l can

be divided out, so that the solution does not depend on l ; this is as it must be, as without amagnetic field nothing distinguishes the l -thick middle layer from the top layer. In addition, inthe infinitely thin layer limit exp(2 kl ) 1 and the solution reduces to the nonmagnetized case.

3. EFFECT ON THE RAYLEIGH-TAYLOR INSTABILITY

If we consider the "pure" Rayleigh-Taylor instability, with no horizontal shear, then U 0

and we are left with

This equation relates the growth rate n to two other inverse timescales on the scale of k -1 an

Alfvn frequency = k 2v and a gravitational timescale = At gk . Expressing thegrowth rate and Alfvn frequency in units of the inverse gravitational timescale and ignoring

the trivial Alfvn wave solution n = A, the result is a quadratic in n2,

To consider the degree to which the magnetized layer stabilizes against the Rayleigh-Taylorinstability, we consider (for simplicity) the maximally unstable case where At 1. Forstability, it is necessary and sufficient that the roots of the quadratic in n2 be positive and real;a quadratic of the form ax2 + bx + c = 0 has positive and real roots for c/a > 0, b/a < 0, and b2 -4ac > 0. The first two of these conditions reduce to

and the third is satisfied for all real . Of these conditions, the first controls, as the second

simply posts a lower limit for of between 2/3 and 1, while the lower limit for the first is

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always greater than 1. Thus, the condition for stability in the case of a Rayleigh-Taylorinstability with At = 1, which would otherwise always be unstable, is

with no strength of magnetic field able to stabilize in the case of an infinitely thin layer. Thisstability criterion is shown in Figure 2 .

(29 kB)

Fig. 2 Rayleigh-Taylor stability boundary for the most unstable (At = 1) case, as given in eq. (34) . The magnetic field strength in the magnetized layer (here expressed as theAlfvn speed squared in units of g /k ) necessary to stabilize against a mode is plotted as afunction of the thickness of the layer in units of the wavelength of the mode. Thus, to

stabilize modes with wavelength

more than an

order of magnitude greater

than the

thickness of the layer l (l / 0.1), a magnetic field such that v 2 g /k is necessary.

4. EFFECT ON THE KELVIN-HELMHOLTZ INSTABILITY

The shear terms in the case of the Kelvin-Helmholtz instability make the dispersion

relation significantly more complicated,

For considering the stability boundary we again consider the most unstable case, in this caseAt = 0. As in the previous case, the growth rate and the Alfvn frequency can be expressed interms of the other timescale of the problem, the advection time across the wavelength of the

perturbation, U = 1/(kU ), leaving us with

While this is a fairly unpleasant quartic in n, it is a fairly approachable quadratic in ,

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The two roots are

For stability, we consider the neighborhood around Im( ) = 0. In this case, both these

branches have minima for purely oscillatory modes around = 1, but in this neighborhood,

the second, negative branch has no real solutions for A with Im( ) 0, so it cannot be

relevant to the question of stability. The positive branch has a minimum of

so that stability is ensured when this condition is met or

which is the same condition for stability of the Rayleigh-Taylor instability, but with 2 U replacing g /k ; recall, however, that the two conditions are for two different values of theAtwood number, such that the instability is maximized in each case. This stability criterion is

plotted in Figure 3 .

(28 kB)

Fig. 3 Kelvin-Helmholtz stability boundary for the most unstable (At = 1) case, asgiven in eq. (40) . The magnetic field strength in the magnetized layer (here expressed as

the Alfvn speed squared in units of the half-shear velocity squared, U 2) necessary tostabilize against a mode is plotted as a function of the thickness of the layer in units of

the wavelength of the mode. Thus, to stabilize modes with wavelength more than anorder of magnitude greater than the thickness of the layer l (l / 0.1), a magnetic field

such that v 2(2 U )2 is necessary.

5. NUMERICAL RESULTS

To confirm the results of the previous sections, numerical experiments were performed intwo dimensions using version 3.0 of the Athena code (Gardiner & Stone 2005 ), adimensionally unsplit, highly configurable MHD code. For the results in this work we used

the

ideal gas MHD solver

with an adiabatic equation

of state ( =

1.4) and the third-order

accurate solver using a Roe-type flux function. We considered a domain size, in code units, of

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[-1/6,1/6] [-1/6,1/6] with resolution 400 400 for the Kelvin-Helmholtz instability

simulations and with a slightly vertically extended domain ([-1/6,1/6] [-1/4,1/4], 400 600)for the Rayleigh-Taylor instability experiments.

To ensure that the resolution used was adequate, a resolution study was performed on a

fiducial run (a Rayleigh-Taylor simulation with B x,0

= 0.07 and vA

= 0.0495; the magnetic

field

code units are such that the Alfvn speed, v = B2/2) with resolution varying between a factorof 2 less than this resolution and a factor of 2 more; measured growth rates varied by only

approximately 3%.

The analytic results presented in previous sections were in the incompressible limit. In ournumerical experiments here, the fiducial density was 1 in code units and the pressure was setso that the sound speed c s would be 1 in code units, which is an order of magnitude larger thanthe velocities achieved in either set of simulations; thus, the Mach number < 0.1, andincompressibility remains a reasonable approximation. In both sets of simulations, amagnetized layer of thickness 1/60 was initialized starting at y = 0, with the strength of themagnetic field varied from run to run. To keep the initial conditions in pressure equilibrium,the thermal pressure was reduced in this layer, but because of the large sound speed (andconsequently large plasma ) this was a small reduction (never more than a few percent).

In both sets of simulations, the interface was given a sinusoidal velocity with a wavelengthequal to the size of the box and an amplitude of v pert 0.0025. In the Kelvin-Helmholtz

case, a background shear velocity of -0.1 in the x-direction was applied to the top layer and themagnetized layer and of +0.1 in the bottom layer. In the Rayleigh-Taylor case, a gravitationalacceleration of g = 0.1 was applied in the negative y-direction. Both sets of simulations usedan Atwood number At = 1/3, so that the top layer had density t = 2. Snapshots of the

evolution of the simulations are shown in Figures 4 and 5.

(107 kB)

Fig. 4 Snapshots from representative runs of the Rayleigh-Taylor instability with a magnetized layer, performed with the Athena code. Shading represents density, andcontours indicate magnetic field strength. In code units, the density of the top fluid is t = 2 and that of the bottom is t

= 1 (At = 1/3), and pressure is set such that in the bottomlayer the adiabatic sound speed c s

is 10, and the domain is in total pressure equilibrium.The wavelength of the initial perturbation is 1/3, and the thickness of the magnetized

layer is 1/60; the acceleration due to gravity g is 1/10 and in the negative y-direction.

Shown are runs with varying initial horizontal magnetic fields, with, from top to bottom, B x,0 =

1/200, 1/20, and 1/10. Snapshots are shown at times in code units of, left to right,

approximately 1.75, 3.5, and

5.25. The contours are

for magnetic field strength

of B x,0/4and 3 B x,0/4.

(109 kB)

Fig. 5 Same as Fig. 4 , but for the Kelvin-Helmholtz instability with the pressure setsuch that in the bottom layer the adiabatic sound speed c s is 1.

Shown are runs with

varying initial horizontal magnetic fields, with, from top to bottom, B x,0 = 0.1, 0.2, and

0.3. Snapshots are shown at times in code units of, left to right, approximately 0.75, 1.5,and 2.25. The contours are for magnetic field strength of B x,0/4 and 3 B x,0/4.

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From the outputs of the simulations, growth rates for the instabilities were calculated byconsidering the growth of the amplitude (measured by finding the mean vertical position ofthe magnetized layer and fitting to a sinusoid of the wavelength of the perturbed mode) as a

function of time. Exponentials were fit to this series of amplitudes, for those times where the

amplitude was resolved by at least three zones and where the amplitude was less than 1/15 of

the wavelength (e.g., before nonlinear evolution begins

to matter). For the

simulations

reported here, this means the fit was performed with amplitudes in the range [1/400,1/45],covering approximately a decade in amplitude. Plotted in Figure 6 are four examples of this

procedure for the Rayleigh-Taylor simulations. For comparison, the results of a resolutionstudy of a fiducial Rayleigh-Taylor case are shown in Figure 7 .

(47 kB)

Fig. 6 Evolution of amplitude of the Rayleigh-Taylor perturbation with time, for several runs (plotted are, top to bottom, B0

= 0.005, 0.05, 0.1, and 0.2), shown as

symbols; also plotted here are best-fit exponentials to measure the growth rate of theinstability. Exponentials were fit to data from those times where the amplitude wasresolved by at least three zones and where the amplitude was less than 1/15 of the

wavelength (e.g., before nonlinear evolution begins to matter); for the simulationsreported here, this means the fit was performed with amplitudes in the range[1/400,1/45], covering approximately a decade in amplitude. The range considered isindicated by thick shaded lines; below the bottom line the "jumpiness" due to poor

resolution on the grid is visible, and the above deviations from exponential behavior dueto nonlinear effects become evident.

(37 kB)

Fig. 7 Same as Fig. 6 , but with B0 = 0.07 and varying the resolution. Shown are severalruns with N varying by a factor of 4, where N is the horizontal resolution in thesimulation, with vertical resolution kept proportionate; thus, N is the number of pointsused to represent one wavelength, with the thickness of the layer being represented by

N /20 points. N = 100 is not quite enough resolution to accurately measure the growth

rate, but the difference

in fitted exponential growth

rates between 200, 400,

600, and800 is only a few percent.

Because of the low speeds of these flows, many time steps (typically on order 20,000) must be taken to evolve these instabilities. Over that period of time, modest amounts of numericaldiffusion slightly modify the structure of the magnetic layer, as does the still slightly

compressible flow itself; the change is shown in Figure 8 . The growth rate of the instabilitiesis very sensitive to the thickness of the layer, and this modification of the magnetic layerthickness must be taken into account when comparing with analytic results.

(40 kB)

Fig. 8 Magnetic field profile through x = 0 for the B x,0 = 1/10 Rayleigh-Taylorsimulation at time 0 ( filled squares ) and 3.75 ( open circles ). The later time profile isshifted leftward by y = 0.00675 for comparison. During the course of the simulation,the thickness of the magnetic field layer at its peak is reduced to about 18 grid pointsfrom the original 20. Besides the inevitable effects of numerical diffusion slightlysmearing out the profile over approximately 18,000 time steps, effects from the slightcompressibility of the flow and asymmetry from the direction of gravity (to the left inthis plot) are shown by the slight increase in magnetic field density and the slope at latetimes.

The comparison of measured growth rates to analytic results is shown in Figures 9 and 10, with theoretical results taken from the numerical solution of equations (30) and (35) . Wherethe numerical experiments can give clean growth rate measurements, the analytic and

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numerical results agree to within a few percent, and there is also agreement at the few percentlevel on the boundary of stability.

(35 kB)

Fig. 9 Rayleigh-Taylor growth rates as a function of magnetic field strength (givenhere in terms of Alfvn speed, vA) in the magnetized

layer, measured from simulation

(diamonds ) and predicted from analytic theory ( lines ). The dashed line gives the

prediction using the nominal width of the magnetized field layer, and the solid linegives the prediction for the actual width of the magnetized layer in the simulations afterdiffusion eats away at the profile as shown in Fig. 8 . The gray point indicates a run doneat twice the resolution to check to see if the default resolution was adequate. Thetheoretical curves are taken from the numerical solution of the dispersion relation, eq.(30) .

(33 kB)

Fig. 10 Same as Fig. 9 , but for Kelvin-Helmholtz growth rates. Below vA 0.1,smaller wavelength modes, seeded by numerical grid noise, grow very quickly andcontaminate any attempt to measure a linear growth rate for the mode underconsideration. The theoretical curves are taken from the numerical solution of thedispersion relation, eq. (35) .

6. DISCUSSION

We have shown, through derivation of dispersion relations and two-dimensional numericalexperiments, that it is possible for even thin magnetized layers to suppress instability growthon scales much larger than their own thickness as long as the magnetic field strength is highenough that the Alfvn velocity in the layer in the direction of the perturbation is of order the

relevant destabilizing velocity scales. In the case of the Kelvin-Helmholtz instability, the mostrelevant case for core mergers, a magnetized layer can stabilize modes an order of magnitude

larger than the thickness of the layer if the Alfvn speed of the same magnitude as the fullshear velocity vA 2U . But

this is almost automatically true; as shown in Lyutikov (2006 ),near the stagnation line the magnetic pressure reaches equipartition with the ram pressure,meaning that this condition on the velocities is met. Thus, we would expect, certainly near thestagnation line, that the magnetic draping "protects" a merging core from instabilities, as

would appear to be the case in simulations published in the literature (Asai et al. 2004 , 2005 , 2006 ; J. Dursi & C. Pfrommer, in preparation).

Clearly, this stabilization only applies to perturbations along the field; but in this plane,then, the draped field can have the twin effects of protecting the merging core against thermal

disruption and

reducing the shear effects

which would tend to

mix in the core

material earlier.In the plane perpendicular to the magnetic field, shear-driven instability will occurunimpeded, leading to a distinct asymmetry in the resulting magnetic layer and, presumably,the moving core or bubble. Such asymmetries have been seen in three-dimensionalsimulations in the literature, as cited above, and would potentially be observable. It isinteresting to note however that even a quite weak field can effect global mixing in the

presence of similar instabilities (e.g., Gardiner & Stone 2007 ), and so even in the plane

perpendicular to the field, it is possible that a thin magnetized layer could keep an interface

sharper than could exist absent the magnetic field.

The full three-dimensional stability problem remains to be tackled, and less clear still is the

effect of a more realistic magnetic field, not expected to be planar or uniform, and the effects

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of fully three-dimensional perturbations on such a layer. Consideration of this morecomplicated and realistic case is left to future work.

L. J. D. is grateful for discussions with M. Ruszkowski, K. Subramanian, C. Pfrommer, andM. Lyutikov, which greatly contributed to this work, for helpful comments on the manuscript

by C. Pfrommer, A. Calder, and M. Zingale,

and for the close

reading and helpful suggestions

and corrections made by the anonymous referee. The author acknowledges funding from the National Science and Engineering Research Council, the hospitality of W. Hillebrandt at theMax-Planck-Institut fr Astrophysik during the beginning of this work, and the hospitality ofthe Kavli Institute for Theoretical Physics during its completion, during which this research

was supported in part by the National Science Foundation under grant PHY 05-51164. Allcomputations were performed on CITA's McKenzie and Sunnyvale clusters which are funded

by the Canada Foundation for Innovation, the Ontario Innovation Trust, and the OntarioResearch Fund for Research Infrastructure. Simulations were performed with version 3.0 ofthe Athena code. Most of the grungy algebra was done with Mathematica. This work madeuse of NASA's Astrophysical Data System.

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